MIDDLEMEN MARGINS AND
GLOBALIZATION1
Pranab Bardhan2, Dilip Mookherjee3 and Masatoshi
Tsumagari4
This version: June 26, 2009
Abstract
We develop a theory of trading middlemen or entrepreneurs who
finance and market goods produced by workers. Brand-name reputations are necessary to overcome product quality moral hazard problems; middlemen margins represent reputational incentive rents. We
develop a two sector two country model of competitive equilibrium,
with endogenous sorting of agents with heterogenous entrepreneurial
abilities into sectors and occupations. Trade liberalization raises inequality between workers and middlemen in the South (which has a
comparative advantage in the more labor intensive good, also subject
to a greater moral hazard problem). Wages remain lower in the South
despite free trade, creating an incentive for Northern middlemen to
offshore production to Southern workers. The distributive effects of
such offshoring tend to be the opposite of trade liberalization.
Keywords: middlemen, reputation, inequality, trade liberalization, NorthSouth trade; JEL Classification Nos: D33, F12, F13, F21
1
Mookherjee’s research was supported by National Science Foundation grant SES0617874. We thank Stefania Garetto, Gordon Hanson, Eric Verhoogen and seminar participants at Namur, IFPRI and Yale for useful comments.
2
Department of Economics, University of California, Berkeley, CA 94720; bardhan@econ.berkeley.edu
3
Department of Economics, Boston University, 270 Bay State Road, Boston MA 02215;
dilipm@bu.edu
4
Department of Economics, Keio University, 2-15-45 Mita Minato-ku Tokyo;
tsuma@econ.keio.ac.jp
1
1
Introduction
Most trade theory is based on production cost differences across countries.
The role of endogenous non-production costs and intermediary markups has
not received comparable emphasis. This paper presents a model of middlemen in international trade. In our model, middlemen play two roles: (a)
value-added managerial activities essential to organize production and transform produced goods into marketable commodities; and (b) brand-name reputations, necessary to assure customers about product quality. Management
activities include financing, product design, contracting with workers and
product marketing. Brand-name reputations are important in overcoming
moral hazard problems in product quality, which cannot be directly observed
by customers at the point of sale. Our model relies on heterogeneity among
agents in the economy with regard to underlying managerial skill. The theory applies to entrepreneurs who supply skilled labor and employ unskilled
workers, as well as to middlemen who operate as intermediaries between customers and producers. In what follows, we interchangeably refer to these
agents as entrepreneurs, managers and middlemen.
We postulate the production function for any good to depend on amounts
of unskilled labor hired by entrepreneurs, and the extent of ‘skilled’ labor
provided by the entrepreneurs themselves. The latter is a shorthand for
the value-added management activities self-supplied by the entrepreneurs to
their own businesses, as in the Lucas (1978) theory of size distribution of firms
based on varying abilities of managers. An alternative interpretation is the
effect of capital market imperfections which create borrowing constraints, as
in theories of occupational choice of Banerjee and Newman (1993) or Galor
and Zeira (1993). In this interpretation, the entrepreneur’s ‘skill’ reflects
the wealth of the entrepreneur, which affects the cost of financing the firm’s
operations since wealthier entrepreneurs have access to cheaper finance when
capital markets are imperfect. Entrepreneurial rents or ‘middleman margins’
reflect returns to these value-added activities as well as reputational rents.
We are motivated by recent empirical estimates concerning the large proportion of consumer prices accounted by middlemen margins for goods exported by developing to developed countries. Feenstra (1998) provides the
following illustration: a Barbie doll sold to US customers for $10 returns
35 cents to Chinese labor, 65 cents covers the cost of materials, and $1 for
transportation, profits and overhead in Hong Kong. Mattel, the US retailer
earns at least $1 per doll. The remaining covers transportation, marketing,
2
wholesaling and retailing in the U.S. Feenstra and Hanson (2004) calculate
Hong Kong markups on re-exports of Chinese goods at 12% of its GDP, while
manufacturing accounted for only 6%. The average markup rate accruing to
Hong Kong intermediaries on re-exported Chinese goods was 24%. Morisset
(1998) reports that the price of coffee declined 18% on world markets but increased 240% for consumers in the US between 1975–93. The average margin
between US consumer price and world price for beef, coffee, oil, rice, sugar
and wheat increased by 83% between 1975–94.
We construct a two sector, two occupation, two country (North-South)
general equilibrium model with endogenous reputational rents earned by entrepreneurs. It departs from most existing literature on trade in two ways.5
First, entrepreneurial rents arise as a result of scarcity of managerial ability, or of wealth (combined with capital market imperfections). Each firm
is identified with a single entrepreneur; the free entry condition takes the
form of equality of firm profits with the rents of the entrepreneur. Only
entrepreneurs with the ability (or wealth) required to manage (or finance)
firms above an endogenously determined threshold actually enter.
The minimum size thresholds for entry depend on the other distinguishing
feature of our model: the need for firms to develop reputations for (unobservable) product quality. In contrast, most existing models of trade and
product quality assume that the latter is observable by customers at the
point of sale (e.g., Auer and Chaney (2009), Hallak and Sivadasan (2009),
Sutton (2007)).6 Problems with unobservable quality have been prominent
in recent scandals concerning quality of Chinese exports of farm products
and toys. We view an essential role of trade intermediaries or entrepreneurs
is to develop and maintain reputation for quality; the margins that accrue
to them represent rents that provide them with the necessary incentives.
We model quality moral hazard problems similar to Biglaiser and Friedman (1994): entrepreneurs have an incentive to make short-term gains from
skimping on quality, which customers will not be able to recognize at the
point of sale, but only afterwards when they consume the good. Reputational considerations naturally create economies of scale: high-volume sellers
5
An exception is the model of Manasse and Turrini (2001), in which entrepreneurs
earn rents owing to a scarcity of skills. Our model differs from theirs by incorporating
unobserved product quality and related reputational phenomena.
6
Bardhan and Kletzer (1984) present a model of choice of technology in a developing
country between a local low-quality process and an imported high-quality process, but do
not embed this in a open economy general equilibrium model.
3
have more at stake if their reputations are destroyed. This is the source of
the minimum size threshold required for entry into the intermediary trade:
we extend the Biglaiser-Friedman model to incorporate these factors.
Given an underlying distribution of entrepreneurial ability or wealth in
the population, agents sort into occupations (entrepreneurship and production workers, or traders and producers) and sectors. The two products differ
in their requirements of labor vis-a-vis value-added activities provided by
entrepreneurs, and in the intensity of the quality moral hazard problem. We
refer to the inputs provided by the entrepreneur and workers as ‘skilled’ and
‘unskilled’ labor respectively. The South has a comparative advantage in the
less skill intensive good (which we denote as the L-good, while the more skill
intensive good is called the C-good). The main assumption on the technology
is that the L-good is subject to a greater moral hazard problem. This reflects greater difficulties in quality inspection and in the difficulty of offering
product warranties owing to lack of durability of the good. The C-good is
more durable; it is produced in a more automated and regulated production
process which is easier to inspect, and thus allows less scope for skimping on
labor or other essential raw material requirements.7
Equilibria in a closed economy are shown to be either of two types, depending on underlying parameters. In one, with a relatively low price of the
L-good, returns to entrepreneurial ability are equalized across the two sectors. This equilibrium is of the traditional “neoclassical” type, where incentive constraints do not restrict the movement of entrepreneurs across sectors.
In the other with a relatively high price of the L-good, the returns to ability
are higher in the L-sector sector. Entrepreneurs would thus strictly prefer to
locate in this sector, but the incentive constraint binds, leading to a higher
minimum ability threshold for entering this sector. Those with high enough
ability enter the L-sector; others with intermediate ability do not meet the
minimum threshold and so have no option but to enter the C-sector instead.8
Equilibria of this latter sort are “non-neoclassical” and exhibit restrictions
7
An offsetting factor would be the greater technological complexity of these goods,
combining a larger number of components in the production process. This may lead
to high costs of ensuring high quality, as emphasized in the O-ring theory of Kremer
(1993). We are assuming that greater difficulties of production inspection and provision
of warranties in farm products or garments outweigh the costs of coordinating a large
number of inputs in high-tech products.
8
Those who do not meet the lower threshold of the C-sector sector become unskilled
workers.
4
on movement of entrepreneurs across sectors owing to the severity of moral
hazard problems associated with entrepreneurship in the lucrative L-sector.
Of particular interest are the distributive effects of trade liberalization in
this model. The relative abundance of unskilled labor in the South confers to
it a comparative advantage in production of the L-good. Trade liberalization
thus raises demand for the L-good in the South. If the economy was operating
in the “non-neoclassical” type of equilibrium, the skill premium (interpreted
as ratio of entrepreneurial rents to wages of workers) must rise within this
sector. Since the incentive constraint associated with entrepreneurship is
binding in this type of equilibrium, expanded production of the L-good requires entry thresholds to be lowered in this sector. This requires reputation
rents earned by L-sector entrepreneurs to rise relative to wages of production
workers (since unskilled wages form the outside option of entrepreneurs, in
the event that they lose their brand-name reputation). On the other hand,
rents of entrepreneurs (hence the skill premium) in the C-sector fall. The
effect on the economy-wide skill premium thus depends on the relative size of
the two sectors: it will rise if the economy operates in the ‘non-neoclassical
regime’ and the L-sector is sufficiently large relative to the C-sector, and fall
if either of these conditions are not met.9
Empirical studies have not found any systematic tendency for trade liberalization to lower or raise skill premia in LDCs, with increased skill premia
found in many instances (Hanson and Harrison (1999), Winters, McCulloch
and McKay (2004), Goldberg and Pavcnik (2007)). Our model provides detailed predictions concerning parameters that influence the trade effect on
skill premia, which could potentially be empirically tested in future research.
For instance, the model provides predictions concerning how the effects vary
across sectors and countries, and how they depend on the underlying distribution of entrepreneurial skill in the economy.
Efficiency effects of trade liberalization include (apart from standard allocative benefits) effects on the gap between entrepreneurial rents in different
sectors and wages of unskilled workers. Agents on the margin between sectors and occupations are subject to discrete increases in incomes as they are
enabled to move to their preferred options by a relaxation of incentive constraints, if the economy is operating in the “non-neoclassical” region. These
constraints depend on prices; hence the model is characterized by pervasive
pecuniary externalities. In the “non-neoclassical” regime, thus, trade is as9
In the “neoclassical” type of equilibrium, classical Stolper-Samuelson results obtain.
5
sociated with income effects in addition to the traditional allocative effects.
If the moral hazard problem in the C-sector is negligible relative to that in
the L-sector, the income effects expand the welfare gains from trade for the
South country, and lower them for the North country. Starting from autarky,
a small expansion of trade can thus be welfare reducing in the North. The
underlying reason is that reputation rents in the L-sector exceed those in the
C-sector, owing to the greater moral hazard problem in the former. These
rent effects swamp the traditional allocative benefits associated with trade
liberalization, since the latter are second-order starting from an autarky position.
The model shows trade liberalization causes a narrowing of North-South
differentials of wages of workers, but does not yield full convergence in the
absence of any trade barriers. This creates an incentive for Northern entrepreneurs to shift production operations to the South. The distributive
implications of such offshoring tends to be the opposite of trade liberalization: e.g., it tends to reduce skill premia in the L-sector in the South, and
(under additional assumptions) raise wages. Moreover, the distributional
effects of full integration (free trade-cum-costless-offshoring) relative to autarky are in general the opposite of free trade alone.
Alternative theories of outsourcing and effects on inequality have been
provided by Antras, Garicano and Rossi-Hansberg (2006), Feenstra and Hanson (1996) and Kremer and Maskin (2003). Of these, the most closely related
to ours is Antras et.al, who develop a model in which agents of heterogenous
abilities sort into hierarchical teams (which extends Kremer and Maskin’s
model in a variety of directions). Inequality rises in the South in their model
owing to the matching of high ability agents in the South with worker teams
from the North. Our theory differs by including two goods, which allows an
analysis of effects of trade liberalization as well as offshoring. Rents arise
owing to moral hazard and reputational effects, rather than matching of heterogenous agents. Their theory predicts that offshoring raises inequality in
the South, whereas our model shows the effect tends to go the other way.
Section 2 introduces the model of the closed economy. Section 3 describes
the equilibrium of the supply-side, where product prices are taken as given.
Section 4 describes the economy-wide equilibrium, and comparative static
properties. Section 5 then extends it to a two country context and studies
effects of trade liberalization. Section 6 considers incentives and effects of
offshoring. Finally Section 7 concludes.
6
2
2.1
Closed Economy Model
Endowment and Technology
We normalize the size of the population to unity. Each agent is characterized
by a level of entrepreneurial skill a, a nonnegative real variable. A fraction
1 − µ of agents have no skill at all: a = 0: we refer to them as unskilled.
The remaining fraction µ are skilled; the distribution of skill isRgiven by a
∞
cdf G(a) on (0, ∞). We shall frequently use the notation d(a) ≡ a ãdG(ã).
The cdf G will be assumed to have a density g which is positive-valued. Then
d is a strictly decreasing and differentiable function.
There are two products L and C that are produced and consumed in
the economy. We now describe the production technology. A product is
produced and sold by an entrepreneur, upon hiring unskilled workers. The
level of production depends on the amount of unskilled labor, and the entrepreneur’s skill. It also depends on the quality of the product, which can
be either high or low. An entrepreneur with skill a employing N unskilled
labor in sector C can produce and sell XC = AC FC (N, a) units of the high
quality good C, and XC = AC FC (N, zC a) units of the low quality good C,
where zC > 1 is a given parameter. Here FC is a CRS, smooth and concave
production function, and AC is a TFP parameter. The entrepreneur may be
tempted to producing the low quality good, in order to produce and sell more
at the same prices. For instance, producing a higher quality product may
demand more supervision or resources provided by the entrepreneur (such as
ability to finance working capital requirements). The goods are experience
goods, with customers unable to observe quality at the time of purchase.
The parameter zC − 1 measures the proportion by which quantity can be
increased by compromising quality, so it represents the severity of the moral
hazard problem in sector C.
Sector L has an analogous technology, described by production levels
XL = AL FL (N, a) for high quality, and XL = AL FL (N, zL a) for low quality,
with zL > 1.
One interpretation of entrepreneurial ‘skill’ is access to finance, if capital
markets are imperfect. The latter implies wealthier agents are less subject to
credit rationing, so ‘skill’ can be interpreted as wealth of the entrepreneur.
Production may require a combination of produced inputs and unskilled labor, and entrepreneurial wealth defines the amount of produced inputs that
can be utilized in the production process. In this interpretation, the dis7
tribution of skill in the population reflects the distribution of wealth. Alternatively, entrepreneurial skill refers to management ability, as in Lucas
(1978), which is relevant with regard to supervision and control of workers.
It could also pertain to marketing skill of the entrepreneur. It will turn out
that entrepreneurs with higher skills will in equilibrium be able to produce
larger quantities, so the skill distribution will relate naturally to the size
distribution of firms.
An underlying assumption is that each firm is managed by exactly one
entrepreneur, so the size of the firm is limited by the skill of its manager.
In practice, of course, firms may employ more than one manager in order
to grow, but problems of managerial moral hazard and coordination across
managers eventually limit the size of firms (as emphasized by a large literature on ‘organizational diseconomies of scale’, e.g., Williamson (1967), Calvo
and Wellisz (1978), Keren and Levhari (1983), Qian (1994) or van Zandt
and Radner (2001)). In order to explore the industry or general equilibrium
implications of limits to the size of firms created by such problems, we adopt
the simplifying assumption that a firm is managed by a single entrepreneur,
similar to Lucas (1978).
Note that neither unskilled or skilled labor is specific to either sector.
Skilled labor is equated with entrepreneurship. Given the assumption that a
firm can have only one manager, there will be no market for entrepreneurs:
every entrepreneur works for herself, i.e., managing her own firm. Entrepreneurial rents will correspond to imputed prices of ‘skill’ which will
be equalized across all agents, and which will clear the market for skill.
Consider the cost-minimizing factor combinations in each sector, when
C
skill is imputed a cost γ relative to unskilled labor: (θN
(γ), θaC (γ)) and
L
(θN
(γ), θaL (γ)) are defined as
C
C
C
(θN
(γ), θaC (γ)) ≡ arg min{θN
+ γθaC | FC (θN
, θaC ) = 1}
and
L
L
L
(θN
(γ), θaL (γ)) ≡ arg min{θN
+ γC θaL | FL (θN
, θaL ) = 1}
The following assumption states that good L is more labor intensive than
good C: for any common ratio of factor costs, production of L uses a higher
ratio of unskilled labor to skilled labor in the cost-minimizing factor choice.
One can think of L as corresponding to agricultural products or low-end
manufactured goods, while C corresponds to high-tech manufactured goods
or services.
8
Assumption 1 For any γ > 0,
C
L
θN
(γ)
θN
(γ)
>
L
C
θa (γ)
θa (γ)
2.2
Entrepreneur’s Profit Maximization and Incentive
Constraint
Profit maximization
Consider an entrepreneur in sector L, facing a product price of pL (with
the C good acting as numeraire, so pC ≡ 1). Suppose the wage rate for
unskilled labor is w. If this entrepreneur were to decide to produce the high
quality version of product L, she would solve the following problem:
max pL AL FL (NL , a) − wNL .
NL
(1)
The optimal employment of unskilled labor NL∗ is a function of pL /w, characterized by the first-order condition
(pL /w)AL ∂FL (NL∗ , a)/∂NL = 1.
(2)
Let Π∗L (pL , w; a) denote the resulting level of profit earned by the entrepreneur. Define
Π∗L
(3)
γL ≡
wa
which can be interpreted as a skill premium in the L sector: the per-skill-unit
return to entrepreneurs relative to the unskilled wage. This skill premium
will play a key role in the analysis below.
L
Recall the definition of (θN
(γ), θaL (γ)) as cost-minimizing factor combination in sector L corresponding to AL unit of production and γ the relative
cost of skill. Given the CRS property of the production function, it follows
9
that10
NL∗
a
,
)
(4)
∗
FL (NL , a) FL (NL∗ , a)
Using the definition (3) of γL , we can express the ratio of product price
to wage as
L
(θN
(γL ), θaL (γL )) = (
pL
NL∗ + aγL
=
w
AL FL
1 NL∗
a
=
[
+ γL ]
A L FL
FL
1 L
=
[θ (γL ) + γL θaL (γL )]
AL N
(5)
This equation can be interpreted as expressing equality of prices with unit
costs, using γL as the imputed price of skill in sector L. It expresses a
monotone increasing relation between the ratio pwL and the skill premium in
sector L.
A similar analysis for sector C yields an analogous price-cost relation:
1
θC (γC ) + γC θaC (γC )
= N
w
AC
(6)
where γC denotes the imputed price of skill in sector C. This equation
expresses a monotone decreasing relation between the wage rate w and the
skill premium in sector C.
Dividing (5) by (6), we obtain the following equation which plays an
important role in the equilibrium analysis in later sections:
pL =
10
L
AC θN
(γL ) + γL θaL (γL )
.
C
AL θN
(γC ) + γC θaC (γC )
Euler’s Theorem implies that FL (NL∗ , a) =
γL
=
=
=
∂FL
∗
∂NL NL
+
∂FL
∂a a.
(7)
Hence
pL
∂FL ∗ ∂FL
N∗
AL [
NL +
a] − L
wa
∂NL
∂a
a
pL
∂FL
AL
w
∂a
∗
∂FL (NL
,a)
∂a
∗ ,a)
∂FL (NL
∂NL
upon using the first order condition (2). It follows from this that (NL∗ , a) is a costminimizing factor combination, if we use γL as the (imputed) factor cost of a unit of
entrepreneurial skill relative to unskilled labor.
10
It can be interpreted as equality of relative prices and unit costs of the two
products. Note that the right-hand-side is increasing in the skill premium in
sector L, and decreasing in the skill premium in sector C. Hence (7) expresses
a relation between the skill premia in the two sectors, and the price pL of
product L relative to C. This can be expressed as follows:
γL = λ(γC ; pL ,
AC
).
AL
(8)
For any given product price pL and set of TFP parameters, it expresses a
monotone increasing relation between the skill premia in the two sectors. And
for any given γC and TFP parameters, it expresses a monotone increasing
relation between pL and γL .
Various properties of this relationship between skill premia in the two
sectors will be used subsequently. For now we note one property which plays
an important role in establishing uniqueness of equilibrium.
Lemma 1
dγL
dγC
C
≡ λ1 (γC ; pL , A
) > 1 whenever γL ≥ γC .
AL
Proof of Lemma 1: Implicitly differentiating (7) we obtain
L
γL +
dγL
AL θaC
θN
+ γL θaL θaC
= pL
=
=
C
L
L
C
dγC
AC θa
θN + γC θa θa
γC +
L
θN
θaL
C
θN
θaC
> 1,
with the second equality using (7), and the last inequality following from
θL
Assumption 1, γL ≥ γC and the fact that θNL is non-decreasing in γL .
a
This Lemma states that if the skill premium in sector L is higher than
in sector C to start with, a further increase in the sector C premium (with
given pL , AL , AC ) must be accompanied by a larger increase in the premium
in sector L. Assumption 1 plays an important role here. Since sector L is
less skill-intensive, an equal increase in the skill premium in the two sectors
will cause unit cost in the L sector to increase by less than in the C sector.
Hence the premium must rise by more in the L sector to ensure that the
ratio of unit costs remains the same.
Incentive Constraint of Entrepreneurs
11
Customers do not observe the quality of the product at the point of sale.
We assume they value only the high quality version of the product, and obtain
no utility from the low quality version. Entrepreneurs will be tempted to
produce the low quality version which enables them to produce and sell more
to unsuspecting customers. The short-run benefits of such opportunism can
be held in check by possible loss of the seller’s reputation. With probability
ηi , an entrepreneur selling a low-quality item in sector i will be publicly
exposed (say by a product inspection agency or by investigating journalists).
In this event the entrepreneur’s brand-name reputation is destroyed, and the
agent in question is forever barred from entrepreneurship. In equilibrium,
customers will purchase only from entrepreneurs for whom the threatened
loss of reputation is sufficient to deter short-term opportunism. Hence in
order for an entrepreneur with skill a to be able to operate in sector i, the
following incentive constraint must be satisfied:
γi wa
w
γi wa
≥ γi wzi a + δ[ηi
+ (1 − ηi )
],
1−δ
1−δ
1−δ
(9)
where δ ∈ (0, 1) denotes a common discount factor for all agents. The lefthand-side of (9) is the present value of producing and selling the high quality
version of good i for ever. The first term on the right-hand-side, γi wzi a represents the short-term profit that can be attained by the entrepreneur upon
deviating to low quality.11 With probability ηi , this deviation results in the
entrepreneur losing her reputation for ever, in which case the agent is forced
to work as an unskilled agent thereafter. With the remaining probability the
entrepreneur’s reputation remains intact.
The incentive constraint can be equivalently expressed as
a ≥ mi /γi
where
mi ≡
(10)
δηi
>1
δηi + (1 − δ)(1 − zi )
is a parameter representing the severity of the moral hazard problem in
sector i. Equation (10) represents a fundamental reputational economy of
scale, which also translates into a sector-specific entry barrier in terms of entrepreneurial skill required. Intuitively, higher skilled entrepreneurs produce
11
Recall that a deviation to low quality is equivalent to an increase in the entrepreneur’s
effective skill from a to zi a.
12
and earn profits at a higher scale (conditional on being able to operate as
an entrepreneur), while the consequences of losing one’s reputation is independent of the level of skill. The stake involved in losing reputation is thus
proportional to the entrepreneurs skill, which has to be large enough for the
agent to be a credible seller of a high-quality good. We assume customers can
infer quality from observing the size of the corresponding firm and existing
prices, by checking whether the incentive constraint is satisfied.
Note that the skill threshold for entry into a particular sector is decreasing in the skill premium in that sector. The reason is simple: a higher skill
premium means entrepreneurs have more to lose from losing their reputations. The skill premia depends in turn on prices and wages; hence so do
the incentive constraints. This will represent the key pecuniary externality
in the model.
Note also that mi > 1 implies that entrepreneurs with skills above the
required threshold for sector i will strictly prefer to be entrepreneurs in sector
i rather than work as an unskilled agent. The per period profit from the
former option is γi wa ≥ wmi > w if (10) is satisfied.
We now introduce the second major assumption of the model, which states
that product L is subject to a more severe moral hazard problem compared
with product C.
Assumption 2 mL > mC
This seems descriptively accurate — agricultural products or low-end
manufacturing products appear to be more prone to quality problems than
high-tech products. This may owe to differences in effective inspection rates,
or to greater scope for manipulating relatively less-automated production
processes. It will also turn out as a consequence that the L sector will be
characterized by higher margins and a more concentrated structure of trade
intermediaries.
3
Industry Equilibrium
In this section we take the product prices as given, and derive the resulting
equilibrium of the supply-side of the economy.
13
3.1
Occupational Choice and Factor-Market Clearing
First we take the skill premia in different sectors as given, and describe occupational choices of agents in the economy. This determines the demand
and supply of unskilled labor. The requirement that the factor market clears
serves to determine a relationship between the skill premia in the two sectors.
Eventually the supply-side equilibrium will be determined by the factor market clearing condition, and the relation (7) between prices and skill premia
in the two sectors.
Occupational choices will depend on skill premia, as these determine both
incentive constraints defining entry into the two production sectors, and their
relative profitability. There are four cases to consider.
mL
Case 1: γL ≥ γC m
C
mL
Since by Assumption 2, m
> 1, it follows that in this case γL > γC also
C
holds. This implies that entrepreneurship in sector L is more profitable than
in sector C. The entry threshold for this sector is also lower, as mγLL < mγCC .
Hence all those with skill above mγLL will enter the L sector, and those below
will become unskilled workers. Clearing of the labor market requires
µ[
L
θN
(γL )
]d(aL ) = µG(aL ) + (1 − µ).
L
θa (γL )
(11)
The production levels will be XC = 0, XL = µAL d(aL )/θaL (γL ).
mL
Case 2: γC < γL < γC m
C
Here γL > γC implies that the L sector is more profitable. On the other hand,
the entry threshold is higher in the L sector: aL = mL /γL > mC /γC = aC .
So agents with a ≥ aL will choose to become L-sector entrepreneurs, while
agents with a ∈ [aC , aL ) are unable to enter the L sector and so have to
be content with becoming C-sector entrepreneurs. And agents with a < aC
become workers.
The labor market clears if
µ[
C
L
mC
θN
(γC )
mL
mC
θN
(γL ) mL
]d(
][d(
)
+
µ[
)
−
d(
)]
=
µG(
) + (1 − µ). (12)
θaL (γL )
γL
θaC (γC )
γC
γL
γC
This provides a condition that relates the skill premia in the two sectors.
It is easily checked that this relationship is downward-sloping, because an
14
increase in the skill premium in either sector tightens the labor market condition. To see this, note that a rise in γC lowers the entry threshold into
the C sector, and raises the demand for unskilled workers for any given C
sector entrepreneur. And on the other hand a rise in the L sector skill premium causes the skill threshold for the L sector to fall, motivating some
entrepreneurs to switch from the C to the L sector. By Assumption 1, and
the fact that γL > γC , L sector entrepreneurs demand more unskilled workers than the C sector entrepreneurs. So the switch of entrepreneurs between
the two sectors tightens the labor market. This is accentuated by the rise in
demand for workers by incumbent L sector entrepreneurs.
In this case, production levels are:
XC = µAC [d(aC ) − d(aL )]/θaC (γC )
XL = µAL d(aL )/θaL (γL )
Case 3: γC = γL
γL = γC = γ, say, implies that entrepreneurs are indifferent between the
two sectors. The L sector is harder to enter, as aL = mL /γ > mC /γ = aC .
Hence agents with a ∈ [aC , aL ) have no option but to enter sector C, while
those with a ≥ aL can enter either of the two sectors. The equilibrium in
this case will involve a fraction of those with a ≥ aL who will go to sector
L, the remaining going to sector C. This fraction must be such as to permit
the factor market to clear. We show now that this in turn translates into an
upper and lower bound for the common skill premium γ.
Denoting the production levels by XL , XC respectively, the factor market
clearing conditions are
L
C
[θN
(γ)/AL ]XL + [θN
(γ)/AC ]XC = µG(aC ) + (1 − µ)
[θaL (γ)/AL ]XL + [θaC (γ)/AC ]XC = µd(aC )
These equations are equivalent to
XL = AL
XC = AC
C
(γ)µd(aC )
θaC (γ)[µG(aC ) + (1 − µ)] − θN
C
L
L
C
θN (γ)θa (γ) − θN (γ)θa (γ)
L
(γ)µd(aC )
−θaL (γ)[µG(aC ) + (1 − µ)] + θN
.
C
L
L
C
θN (γ)θa (γ) − θN (γ)θa (γ)
15
However since only agents with a ≥ aL have the option to become L-sector
entrepreneurs,
XL ≤ µAL d(aL )/θaL (γ)
which implies
µ[
L
C
(γ) mL
θN
(γ)
mC
θN
mC
mL
)] ≥ µG(
]d(
)
+
µ[
][d(
) − d(
) + (1 − µ). (13)
L
C
θa (γ)
γ
θa (γ)
γ
aL
γ
On the other hand, XL ≥ 0 implies
µ[
C
θN
(γ) mC
mC
]d(
)
≤
µG(
) + (1 − µ).
θaC (γ)
γ
γ
(14)
Inequalities (13, 14) provide lower and upper bounds on the common premium rate γ. Note that (13) is the inequality version of the factor market
clearing condition (12) in Case 2. Hence the lower bound in Case 3 exactly
equals the limiting premia in Case 2 as γL and γC approach each other. This
will be shown in the figure below.
Case 4: γC > γL
γC > γL implies that sector C is more profitable. Also the entry threshold
in sector C is lower. In this case no entrepreneur enters sector L. Those
with skill a ≥ aC enter sector C, the rest become workers. The labor market
condition is
θC (γC )
µ[ N
]d(aC ) = µG(aC ) + (1 − µ).
(15)
θaC (γC )
The production levels are
XL = 0
XC = µAC d(aC )/θaC (γC ).
The relationship between skill premia in the two sectors that ensure the
labor market clears is shown in Figure 1 below. The thresholds depicted γC1 ,
γC2 and γC3 are defined by the solutions to the following equations.
µ[
µ[
L mL 1
θN
( mC γC )
1
mL 1 ]d(mC /γC )
L
θa ( mC γC )
= µG(mC /γC1 ) + (1 − µ)
L
C
θN
(γC2 )
θN
(γC2 )
2
]d(m
/γ
)+µ[
][d(mC /γC2 )−d(mL /γC2 )] = µG(mC /γC2 )+(1−µ).
L
C
θaL (γC2 )
θaC (γC2 )
16
µ[
C
θN
(γC3 )
]d(mC /γC3 ) = µG(mC /γC3 ) + (1 − µ)
3
C
θa (γC )
For future reference, we shall denote this relationship by the equation
γL = ψ(γC ).
(16)
Note that this function depends on parameters µ, mL , mC but is independent
of pL or TFP parameters AL , AC . Note also that this function is well defined
for γC < γC3 , and is not a monotone relationship: it is decreasing below γC2 but
increasing thereafter. The downward-sloping part corresponding to Case 2 is
the ‘non-classical’ region where skill premia are not equalized across sectors.
The upward-sloping part corresponding to Case 3 coincides with the line of
equality, so this is the ‘classical’ region where skill premia are equalized. Note
mL
that the greater the relative severity m
of the moral hazard problem in the
C
L-sector, the greater the range occupied by the non-classical region.
Figures 2 and 3 subsequently display changes to this relationship resulting
from an increase in mL and µ. An increase in mL raises entry barriers into
the L-sector, which tends to raise the L-sector premium γL in Cases 1 and
2. A rise in µ raises the fraction of agents with entrepreneurial skills, which
tighten the labor market, raise wages, and lowers skill premia in both sectors.
3.2
Supply-Side Equilibrium
We are now in a position to characterize the supply-side equilibrium, by
putting together the condition that the labor market clears (which incorporates reputation effects, occupational and sectoral choices by entrepreneurs),
with the relation between prices and costs representing profit maximization
by active entrepreneurs in each sector. The former is represented by the
relation between skill premia that clears the factor markets, shown in the
previous section. The latter is represented by the upward-sloping relation
(7) between premia in the two sector for any given product price pL . Geometrically it is represented by the intersection of the corresponding relations
between the two skill premia. This is shown in Figure 4 for different values
of pL .
Proposition 1 For given pL > 0, as well as other parameters µ, AL , AC , zL , zC ,
there is a unique supply-side equilibrium determining skill premia and outputs
of the two sectors, as well as the wage rate.
17
γL =
γL
mL
γ
mC C
γL = γC
γC1
γC3
γC
γC2
Figure 1: Relation Between Skill Premia for Factor Market Clearing
We sketch the proof informally. For i = 1, 2, 3, define threshold prices piL
corresponding to threshold C-sector skill premia defined previously (which
mark the transition between Cases 1,2, 3 and 4):
p1L = (AL /AC )
mL 1 L mL 1
γ θ ( γ )
mC C a mC C
1 C
+ γC θa (γC1 )
L mL 1
θN
( mC γC ) +
p2L = (AL /AC )
C
θN
(γC1 )
L
θN
(γC2 ) + γC2 θaL (γC2 )
C
θN
(γC2 ) + γC2 θaC (γC2 )
L
θN
(γC3 ) + γC3 θaL (γC3 )
= (AL /AC ) C 3
θN (γC ) + γC3 θaC (γC3 )
Now consider the following price ranges.
p3L
Case 1: pL ≥ p1L
In this case, there is an equilibrium with γC ≤ γC1 , with complete specialization in product L, and production levels XL = µd( mγLL )/θaL (γL ), XC = 0.
Since the price-cost relation (7) between skill premia in the two sectors is
upward-sloping, it is evident there cannot be any other equilibrium. In the
interior of this range, the equilibrium outputs are locally independent of pL .
18
γL
γL =
mL
γ
mC C
γL = γC
γC
Figure 2: Effect of a Rise in L-sector Moral Hazard Problem on FactorMarket-Clearing Skill Premia
19
γL
γL =
mL
γ
mC C
γL = γC
γC
Figure 3: Effect of Rise in µ on Factor Market-Clearing Skill Premia
20
γL
p1L
Case 1
p2L
Case 2
p3L
Case 3
Case 4
γL =
mL
γ
mC C
γL = γC
γC1
γC3
γC2
Figure 4: Supply-Side Equilibrium
21
γC
Case 2: p1L > pL ≥ p2L
Here there is an equilibrium corresponding to the downward sloping stretch
in the relation between γC , γL expressing labor market clearing. This follows
from the fact that at p1L there is an equilibrium corresponding to γC1 , and at
p2L there is an equilibrium corresponding to γC2 . Moreover in this case there
cannot be any other equilibrium owing to Lemma 1.12
In the interior of this range of prices, increasing pL results in an increase
in XL and γL , and a decrease in XC and γC .
Case 3: p2L > pL ≥ p3L
Now there will be an equilibrium in which skill premia are equalized across
the two sectors. The same argument as in Case 2 ensures the equilibrium
is unique.13 The equilibrium skill premium in this case γL = γC = γ ∗ is
determined by the condition
pL = (
L
AC θN
(γ ∗ ) + γ ∗ θaL (γ ∗ )
) C ∗
.
AL θN (γ ) + γ ∗ θaC (γ ∗ )
(17)
It is evident that an increase in pL will increase XL , reduce XC and the
common skill premium γ ∗ . The latter results as the shift in production
towards the L sector raises the demand for labor, inducing a rise in the wage
rate.
Case 4: pL < p3L
In this case, there is an unique equilibrium with perfect specialization in
sector C, with production level
XC = µAC d(aC )/θaC (γC ).
An increase in pL in this region will raise γL , while leaving XC , γC unchanged.
12
2
If there were another equilibrium, it would have to lie in the range γC > γC
. But this
would require the slope of the skill premium relationship expressing (7) to have a slope
smaller than one somewhere above the 45 degree line, which is ruled out by Lemma 1.
13
Note in particular that Lemma 1 ensures that the slope of the relation between skill
premia expressing (7) strictly exceeds unity even on the 45 degree line. Hence a tangency
of this relation with the 45 degree line is ruled out.
22
γL∗
γC
p2L
p1L
p3L
Figure 5: Effect of Changes in pL on Skill-Premia
3.3
pL
Comparative Static Properties of the Supply-Side
Equilibrium
The distributional effects of changes in parameters can now be deduced.
3.3.1
Rise in pL
Recall from the price-cost relations (5, 6) that the equilibrium wage rate w
moves inversely with the skill premium in the C sector, and pwL moves in the
same direction as the skill premium in the L sector. And from the discussion
above, we can derive the effect of changing pL on skill premia in the two
sectors. This relation is depicted in Figure 5.
Focusing on the region p1L > pL > p2L which corresponds to Case 2 above,
a rise in pL raises the skill premium in sector L and lowers it in sector C. This
implies that w rises, while pwL rises. Hence the wage rate expressed in units
of the C good rises, but expressed in units of the L good falls. The effect on
utility of workers thus depends on relative preferences in their consumption
for the two goods: if biased in favor of the L good sufficiently, workers will be
worse off. Treating entrepreneurial skill as the other factor, the return to the
23
unskilled factor relative to the skilled factor in this sector wγwL goes down: an
anti-Stolper-Samuelson result. The reason is that an expansion in the output
of this sector is associated with an increase in rents earned by entrepreneurs
in the L sector, and a lowering of entry thresholds into this sector (i.e., a
weakening of the reputational constraints that restrict entry). This requires
entrepreneurial returns to rise relative to worker earnings, as the latter serves
as the punishment payoff associated with a loss of reputation.
Note, however, that over the price range pL ∈ (p3L , p2L ) the StolperSamuelson result does hold. Here skill premia are equalized across the two
sectors, and an increase in pL is associated with a fall in the economy-wide
skill premium, while the wage rate for unskilled workers rise. The behavior
of the economy over this price range conforms to the Heckscher-Ohlin model.
The distributional effects for the other price range (p2L , p1L ) can be shown
in more detail in Figure 6. This shows the distribution of income across agents
with varying skills, and how it changes as a result of an increase in pL . Agents
with skill below the entry threshold aC for the C sector earn the unskilled
wage w. Between the thresholds aC and aL for the two sectors, the agents
are C-sector entrepreneurs, earning γC wa. By definition of the threshold
aC = mγCC , it follows that the earning of a C-sector entrepreneur at this
threshold equals wmC , which strictly exceeds w as mC > 1. Hence there is a
discrete upward jump in earnings at the entry threshold for entrepreneurship.
There is a similar discrete jump in earnings at the threshold aL for entry
into the L sector, owing to the difference in skill premia between the two
sectors. The highest incomes accrue to entrepreneurs in the L-sector, who
manage the largest firms in the economy. They are followed by C-sector
entrepreneurs, who manage smaller firms, and finally workers who work as
unskilled employees in both sectors.
Figure 6 shows how the distribution of income is altered following a rise
in pL : the skill premium in the L-sector rises, while that in the C-sector falls,
and the wage rate rises. For the economy as a whole the change in distribution
of income consists of a rise in incomes at the top and the bottom, and a fall
in incomes in the middle. Within the export (L) sector, however, inequality
rises, and falls within the import (C) sector. Note that these inequality
effects appear within firms.
24
wmL
wγL a
wγC a
wmC
w
a
aL
aC
Figure 6: Income Distribution Changes Resulting from Increase in pL ∈
(p2L , p3L )
25
γL
γC
Figure 7: Effect of Increase in µ on Supply-Side Equilibrium
3.3.2
Effect of TFP Changes
Now consider the effect of a change in TFP parameters AC , AL . These parameters have no effect on the factor market clearing condition. Their only
AL
effect is on the price-cost relation (7): a rise in A
has the same effect as a
C
rise in pL . Hence the distributional effect of changes in relative TFP in the
two sectors is qualitatively the same as the effect of a change in their relative
price.
3.3.3
Effect of Changes in Skill Endowment
Next consider the effect of an increase in µ, the proportion of agents in the
economy with skills. As shown in Figure 3, the frontier between skill premia
corresponding to the factor market-clearing condition (16) shifts inwards,
owing to the resulting tightening of the market for unskilled labor. Excepting
the case that γC = γL is maintained before and after the change in µ, both
skill premia tend to fall, as shown in Figure 7. However, the effect on the
relative production levels XL /XC is complicated, we turn to this now.
Since the C good is more skill-intensive, one would intuitively expect an
increase in endowment of skilled labor in the economy to raise the relative
production of C, as the Rybczynski Theorem potsulates in conventional trade
theory. This indeed is true in the ‘classical’ region with equal skill premia in
the two sectors. From (17) which is independent of µ, it is evident that a rise
26
in µ leaves the skill premium unchanged. Hence the entry thresholds into the
two sectors and the demand for unskilled labor from each active entrepreneur
of the same skill is also unaffected. Since the L sector is less skill-intensive,
it follows that the production of the L good must fall, in order to allow the
market for unskilled labor to clear.
In the ‘non-classical’ region where the skill premium in the L sector is
higher, there will be an additional effect of a change in µ on skill premia
in the two sectors. What turns out to matter is the change in the relative
skill premia in the two sectors. An increase in µ tightens the labor market
and thus tends to drive the unskilled wage higher. Since the L sector is less
skill-intensive, this tends to lower the skill premium in the L sector by more
than in the C sector. However, the relative skill premium γγCL may still go up,
if it was high enough to start with.14 In that case, we obtain a countervailing
XL
effect which can raise X
.
C
To see this concretely, we consider the following example, where the
density of the skill distribution does not fall too fast, and the production
functions for both sectors have constant and equal elasticity of substitution
between skilled and unskilled labor.
2
g(a)
Proposition 2 Assume that ad(a)
is increasing in a and production function
for i = L, C exhibit constant, equal elasticity of substitution κ(≥ 0):
1/κ
κ−1
κ−1
κ
Xi = Ai Fi (Ni , ai ) = Ai (ki ai κ + Ni
κ
) κ−1
(18)
with kC > kL (to ensure Assumption 1 is satisfied). Then in the supply-side
equilibrium:
k
(i) If κ ≥ 1 −
log[ kC ]
L
m
log[ m L
C
for any pL ∈
, an increase in µ has the effect of decreasing XL /XC
]
3
(pL , p1L ).
log[
kC
]
(ii) If 0 ≤ κ < 1− log[ mkLL ] , the increase in µ has the effect of reducing XL /XC
mC
for any p ∈ (p3L , AC /AL ). Moreover, there exists p̄L ∈ (AC /AL , p1L ) such
that XL /XC is increasing in µ for any p > p̄L .
14
Specifically, the tighter labor market tends to lower γC , and the effect on the relative
d(
γL
)
dγL
premium γγCL of a change in γC is dγγCC = γ1C [ dγ
− γγCL ] which is negative if
C
i.e., the initial value of the relative premium is high enough.
27
dγL
dγC
<
γL
γC ,
pL
p1L
AC /AL
µ↑
p2L
p3L
XL /XC
Figure 8: Effect of Increase in µ on
XL
XC
in Supply-Side Equilibrium
The proof of this is provided in the Appendix. The Rybczynski theorem
is valid if the elasticity of substitution is large (case (i)) and otherwise for
C
values of pL below A
, but not for values of pL close enough to p1L . In the
AL
latter case the relative skill premium in the L sector is sufficiently high to
start with that it increases as a result of the increase in endowment of skilled
labor. This is strong enough to cause the relative production of the less skillXL
.
intensive good to rise. Figure 8 provides an illustration of the effect on X
C
In the context of the open economy, this will provide an instance where the
Leontief paradox appears, if a North and South country differ only in their
relative endowments of skilled and unskilled labor.
4
Autarky Equilibrium
Now we close the model of the autarkic economy by specifying the demand
side.
28
4.1
Demand-Supply Condition
There is a representative consumer with a homothetic utility function U =
U (DC , DL ), where DC , DL denote consumption of the two goods. The relative demand function is then given by
DL /DC = φ(pL )
where φ(pL ) is continuous and strictly decreasing in pL . We assume that
limpL →0 φ(pL ) = ∞ and limpL →∞ φ(pL ) = 0.
The economy-wide equilibrium is then represented by equalization of relative supply and relative demand:
DL /DC = φ(pL ) = XL /XC
(19)
XL
on pL is provided by the supplywhere the dependence of relative supply X
C
side equilibrium described in the previous section.
Proposition 3 There is a unique autarkic equilibrium, in which pL ∈ (p3L , p1L ).
This follows from the fact that relative demand is continuous and strictly
decreasing in pL , while relative supply is well-defined for pL ∈ (p3L , p1L ), and
over this range is continuous and strictly increasing in pL . Moreover, as pL
tends to p3L , relative supply of the L good tends to 0 while relative demand is
bounded away from zero. And as pL tends to p1L , relative supply of L tends
to ∞, while relative demand is bounded.
The autarky equilibrium (pL , γL , γC , w) is characterized by conditions of
profit-maximization (5), (7); the labor market clearing condition (16), and
the product-market clearing condition (19). It is illustrated in Figure 9.
4.2
Comparative Static Effect of TFP Changes
Consider the effect of an increase in AC /AL on the autarky equilibrium. As
shown in the previous section, it causes a decrease in XL /XC in the supplyside equilibrium for any pL ∈ (p3L , p1L ). Therefore the autarky price pL will
increase.
However the effect on skill premia γC and γL is more complicated. From
(5) and (7), these are determined by pL AL /AC . If the equilibrium level of
pL AL /AC increases, the skill premia respond the same way as they do in the
supply-side equilibrium when pL increases. Whether this is the case depends
on the elasticity of substitution between the two goods on the demand-side.
29
pL
XL /XC
p∗L
DL /DC
Figure 9: Autarkic Equilibrium
0
Proposition 4 Let κD denote the elasticity −pL φ (pL )/φ(pL ) of relative demand at the autarkic equilibrium price p∗L . Then a small increase in AC /AL
causes p∗L AL /AC to increase if and only if κD < 1.
Proof of Proposition 4
Define p̃ ≡ pL AL /AC . The product market clearing condition can be written
as
φ(p̃AC /AL )AC /AL = (XL /XC )(AC /AL ).
(20)
Then note that from (7) and (16), γL and γC can be expressed as functions of p̃ alone. It is easily checked from the calculation of the supply-side
equilibrium quantities that the right-hand-side of (20) can be expressed as
a function of γL , γC alone. Hence this is a function of p̃, i.e., conditional on
the value of p̃, it is independent of AC /AL . Moreover it is strictly increasing
in p̃ over the relevant range.
Since the left-hand-side of (20) is strictly decreasing in p̃ for any given
value of AC /AL , it follows that the equilibrium value of p̃ is uniquely determined by this equation for any given AC /AL . The left-hand-side is locally
increasing in AC /AL if and only if κD < 1, whence the equilibrium value of
p̃ increases in AC /AL .
30
Hence with Cobb-Douglas utility which is associated with κD = 1, changes
in TFP parameters will have no effect on equilibrium skill premia. With inelastic relative demand, a ceteris paribus rise in TFP in the C-sector will
raise skill premium in the L sector and lower it in the C sector. The profit
maximization conditions (5, 6) then imply that w and pwL will rise. The
distributive consequences will then be similar to those for the supply-side
equilibrium when pL rises, illustrated in Figure 6.
4.3
Comparative Static Effect of Changes in Skill Endowment
Now consider the effect on the autarky equilibrium of increasing µ. Recall
XL
that the effects of this on X
in the supply-side equilibrium were quite comC
plicated. However it turns out that the distributional effect on the autarkic
equilibrium is quite simple: income inequality tends to fall as wages rise,
while skill premia in both sectors fall.
Proposition 5 Consider a small increase in µ. In the autarkic equilibrium
this lowers skill premia in both sectors, while w and pwL both rise.
Proof of Proposition 5
First consider the case where the increase in µ locally reduces XL /XC ,
implying that equilibrium pL rises. Suppose that the initial price level is in
(p2L , p1L ). Then as explained previously, taking pL as given, the increase in µ
has the effect of decreasing γC and γL in the supply-side equilibrium. On the
other hand, the increase in pL causes γC to fall and γL to rise. Therefore the
total effect on γC is negative. Since the equilibrium pL rises, the equilibrium
level of XL /XC becomes lower. However the right hand side of
XL /XC =
d(aL )
AL θaC (γC )
L
AC θa (γL ) [d(aC ) − d(aL )]
increases with a decrease in γC , which implies that the total effect on γL must
be negative. From (5) and (6), the effect on w and w/pL must be positive.
On the other hand, if the price level is in (p3L , p2L ), the increase in µ does not
have a direct effect on γC and γL for given pL , and the effect on both through
the increase in pL is negative.
31
Next, consider the case where the increase of µ increases XL /XC locally
at the initial equilibrium. This is possible only if pL ∈ (p2L , p1L ). Then the
direct effect of µ taking pL as given is negative for both γL and γC . On the
other hand, the indirect effect through the decrease in pL is negative for γL
and positive for γC . Hence the total effect on γL is negative. A symmetric
argument to that in the previous paragraph also implies that the total effect
on γC is negative.
In sum, regardless of the effect of µ on pL , the increase in µ makes γL
and γC go down and w and w/pL go up.
5
Free Trade Equilibrium
We are now in a position to examine the effect of trade between two countries
that differ in endowments or technology. Suppose there are two countries H
and country F , the former corresponding to less developed country. We shall
suppose they differ in either of two parameters: country F has a higher µ
the proportion of skilled agents (µH < µF ), or higher TFP in the C sector
F
(AH
C < AC ). In all other respects the two countries are identical. As discussed
in the previous section, the differences in endowments or technology will
create differences in the relative product price between the two countries in
the absence of any trade.
In the free trade equilibrium (with zero transport costs), there will be a
common equilibrium price pTL in the two countries, determined by
DLH + DLF
XLH + XLF
= H
DCH + DCF
XC + XCF
(21)
where both relative demand and supplies in each country will depend on the
common price. Once pTL is determined, the respective supply-side equilibrium
of each country will determine the remaining variables in each country.
5.1
Difference in Skill Endowment
Suppose µH < µF , and the two countries are the same on all other dimensions. From the discussion in the previous section, the ordering of relative
supplies of the two goods can go either way in autarky. We first consider
the ‘regular’ case where the relative supply of the L good is lower in F , so
32
country H has a comparative advantage in production of the L good. Let
pjA
L denote the autarky price in country j. In most of the discussion that
follows, we shall assume that the equilibrium (both under autarky and free
trade) in both countries falls in the range corresponding to Case 2 where skill
premia in the L sector are higher.
(1) pHA
< pFL A
L
In this case, country H has comparative advantage in the production of
good L. As a result of the free trade, the equilibrium price level pTL will lie
FA
in (pHA
L , pL ), country H will export good L and import good C. This is the
same as in a standard Heckscher-Ohlin model.
The distributional effects are somewhat different, however, as illustrated
in Figure 10. Trade will raise pL in country H, which as discussed previously
will raise the skill premium in the L sector, as well as the wage w. So trade
liberalization tends to be accompanied by higher income inequality within
the export sector of the developing country: increasing exports of the L good
benefits entrepreneurs in the L sectors by more than it benefits unskilled
workers. On the other hand, the skill premium in the C sector will decline,
so the effect on income inequality in the H country as a whole is ambiguous.
See Figure 6 again.
Moreover, factor prices are not equalized as a result of free trade. With a
common price ratio, country F has a higher skill endowment, and a tighter
labor market, so the wage rate is higher in F , and the skill premium in the C
sector is lower in F . Since pL is the same in the two countries, pwL is higher in
H, and the L sector skill premium is also higher. Thus skill premia in both
sectors are higher in H, while the wage rate is lower.
(2) pHA
> pFL A
L
In this case, country H has the comparative advantage in production of
good C. As a result of free trade, the equilibrium price level pTL in the
trade is in (pFL A , pHA
L ), country H exports good C and imports good L. This
corresponds to the Leontief Paradox in Heckscher-Ohlin theory. In this case,
the distributional effects of trade are completely the opposite of the previous
case. But wF T > wHT is maintained.
33
γL
pFL A
HT
FA
FT
pTL
pHA
L
HA
γC
w/pL
wHT
wF T
w
Figure 10: Comparing Free Trade with Autarky: the ‘Regular’ Case
34
5.2
Differences in Technology
F
Now consider the case of AH
C < AC . Recall that in the autarky equilibrium,
an increase in AC raises pL . Hence country H will have a comparative advantage in good L, and pHA
< pFL A . As a result of free trade, the equilibrium
L
T
HA F A
price pL will lie in (pL , pL ). Country H exports good L and imports good
C.
The effect of trade is thus akin to the effect of a rise in pL in country H,
1j
and a fall in country F . If pTL is in (p2j
L , pL ) for j = H, F , in country H the
skill premium in the L sector rises and falls in sector C, the case described in
Figure 6. The opposite happens in country F . Factor prices are not equalized
in the trade equilibrium: γLHT > γLF T , γCHT < γCF T and wHT < wF T .
5.3
Welfare Effects of Trade
The effects of trade on the equilibrium outcomes of each country are represented by the effect of trade on relative product price pL and thereafter
on the resulting supply-side equilibrium. Hence it suffices to examine the
welfare effect of changes in pL , which can be shown to be equivalent to the
following expression.15
Proposition 6 The aggregate welfare effect in country j of a change in pjL
has the same sign as
(XLj − DLj ) + wj µ[(1 − mC )g(ajC )dajC /dpjL − (γLj − γCj )ajL g(ajL )dajL /dpjL ] (22)
In addition to the standard allocative effect XLj − DLj , there is an additional set of welfare effects operating through the change in entry thresholds
aL and aC . This owes to the upward jumps in incomes at these thresholds,
owing to the binding incentive constraints operating at these thresholds. A
relaxation of these thresholds enables agents to transfer occupations (from
being a worker to an entrepreneur, when aC falls) or sectors (from sector
C to sector L, when aL falls) and experience a discrete income gain. This
explains the second and third terms in the expression above. When aC falls,
the change in income is −wj (1 − mC ) for every agent at the threshold, who
is switching from being a worker to a C-sector entrepreneur. When aL falls,
the change in income is proportional to the difference in skill premia between
the two sectors.
15
The Proof is postponed to the Appendix.
35
In general, it is difficult to sign the sum of these income effects resulting
from a change in entry thresholds following a relaxation of trade barriers.
For instance, consider the cases described above where country H has a
comparative advantage in the L good, and it is operating in the regime
where the L sector has a higher skill premium. Trade causes an expansion
in the L sector (a fall in aL ) which tends to raise aggregate entrepreneurial
incomes. It also causes a contraction in the C sector (a rise in aC ) which
tends to reduce aggregate income, as some C-sector entrepreneurs switch to
becoming workers. The net effect is ambiguous. However, in the case where
the moral hazard problem in the C sector is negligible (i.e., mC approaches
one), the pecuniary externality at the aC threshold vanishes. In that case the
aggregate income effect of trade is positive for country H, which adds to the
standard allocative benefits of trade. Conversely, the aggregate income effect
is negative for country F , which subtracts from the allocative benefits. Hence
starting from autarky, a small expansion of trade can be welfare-reducing for
country F ! We summarize this discussion below.
Proposition 7 Suppose country H has a comparative advantage in the L
good, and the moral hazard problem in the C sector is negligible (mC approaches 1), while that in the L sector is non-negligible (mL is bounded away
from 1). Suppose also that both countries operate in the region where skill
premia are not equalized across the two sectors. Then country H (resp. F )
experiences a welfare gain from trade in excess of (resp. less than) the standard allocative benefits. Starting from autarky, a small lowering of trade
barriers will lower welfare in country F .
With different parameter values, these welfare results can get reversed.
For instance, suppose that the moral hazard problem in the C sector is nonnegligible, and approximately the same as in sector L (i.e., mC is bounded
away from 1 and mL − mC is negligible). Then only movements in the entry
threshold aC will generate non-negligible income effects. If country H has
comparative advantage in the L good, trade will generate negative income
effects for country H and positive income effects for country F . Starting
from autarky, small trade expansion will then lower welfare in H, and raise
it in F .
36
6
Offshoring
The analysis in the previous section showed that free trade between H and
F does not lead factor prices to be equalized. In particular, country H with
either a lower endowment of skilled labor or lower TFP in the C sector, will
end up with a lower wage rate for unskilled workers. Hence entrepreneurs
in F will have an incentive to organize their production by hiring workers in
H rather than in F . This section examines the implications of this type of
offshoring. We restrict ourselves to the case where the two countries differ
only in their skill endowment: µF > µH .
In addition to free trade in goods, we suppose that entrepreneurs in both
sectors and countries can costlessly locate their production operations in either country. The following proposition shows that the resulting equilibrium
H
L
,
is identical to that in the completely integrated economy with µG ≡ µ +µ
2
with factor prices equal across the two economies.
Proposition 8 With free trade and costless offshoring, the equilibrium is
equivalent to that in the completely integrated economy with µG proportion
of skilled agents. In this equilibrium, wH = wF . With superscripts A and
G denoting the autarky and the fully integrated equilibrium respectively, if
γLiA > γCiA for i = H, F ,
(i)γLF A < γLG < γLHA
(ii)γCF A < γCG < γCHA
(iii) wF A > wG > wHA
Proof of Proposition 8
Suppose that wH 6= wF with free trade and costless offshoring. If wH < wF ,
all entrepreneurs would hire only workers in country H. However unskilled
workers in country F do not have the option to become entrepreneurs, and
would thus be unemployed, implying wF = 0, a contradiction. Similarly,
we cannot have wH > wF . With a common product price ratio pL and
the common unskilled wage, skill premia must be equalized in each sector
across the two countries. These premia must clear the market for unskilled
workers in the integrated economy, i.e., satisfy (16) with µG representing the
proportion of skilled agents.
37
As shown in the autarky equilibrium, in the region that γL > γC holds in
the equilibrium, the autarky levels of γ C and γ L are decreasing in µ regardless
of its impact on pL . Hence µH < µG < µF implies (i) and (ii). Since γC and
w are inversely related, (iii) follows.
In the integrated equilibrium, the absence of any trade or offshoring costs
implies that entrepreneurs are indifferent which country to locate their operations. This implies that the structure of trade is indeterminate. This
indeterminacy could however be resolved by the presence of small trading
and offshoring costs. In either case, the difference between the total size of
workers hired by country H’s entrepreneurs and the total size of H’s workers
must be lower than that in country F , since µF > µH . In other words, the
net size of the outsourcing must be larger for country F ’s entrepreneurs.
The effects of full integration relative to autarky are shown in Figure 11.
Proposition 8 indicates that the distributional effect of full integration differs
considerably from that of free trade alone. The latter tends to raise the skill
premium in the L sector in country H, implying L-sector entrepreneurs gain
more than do unskilled workers. Integration on the other hand causes the skill
premium in country H to fall (as indicated by (i)), while the unskilled wage
rises (indicated by (iii)). Hence inequality between workers and entrepreneurs
within the export sector falls as a result of integration, in contrast to the
effects of trade liberalization alone.
The comparison of full integration with free trade is somewhat more complicated. Suppose that under autarky country H has comparative advantage
in the L good. Since γLHT > γLHA > γLF A > γLF T , it follows that
γLHT > γLG > γLF T ,
i.e., offshoring causes skill premia in sector L to fall in country H, and rise
in country F . This is intuitive: the higher skill premia in this sector under
free trade in country H owe to the lower unskilled wage. Offshoring allows
country F entrepreneurs to employ unskilled workers in country H, erasing
the disadvantage they had vis-a-vis country H entrepreneurs.
However the effect of offshoring on γC and w is not so clear. If relative
product prices were to remain unchanged, then offshoring would raise the
unskilled wage and lower the C-sector skill premium in country H. In that
case, offshoring unambiguously tends to reduce income inequality between
38
γL
µ
G
µH
pFL A
pG
L
pHA
L
γLHA
γLG
γLF A
µF
γC
γCF A γCHA
γCG
Figure 11: Comparison of Autarky with Full Integration
39
unskilled and skilled agents in country H. However, offshoring could have
another effect, by causing a change in pL . This effect is not easy to sign,
however. If terms-of-trade effects of offshoring are insignificant we can infer
that it tends to generally improve income distribution in favor of unskilled
workers in both sectors in country H.
7
Concluding Comments
We have constructed a theory of entrepreneurial or intermediary rents in a
general equilibrium model of trade. These rents arise as return to key roles
in financing and marketing they play. Of particular importance is the role
of their brand-name reputations which are necessary to overcome product
quality moral hazard problems vis-a-vis customers. The model helps explain
why trade liberalization can increase inequality of earnings between production workers and entrepreneurs in developing countries. Reputational factors
and limits to the size of firms reflecting fundamental moral hazard problems
with regard to entrepreneurial functions such as management, finance and
marketing prevent costless reallocation of factors across sectors, resulting in
rising inequality within sectors and firms. The model explains incentives for
Northern countries to offshore their production to Southern countries, and
predicts the distributive implications of such offshoring to be the opposite of
trade liberalization.
Apart from margins earned by intermediaries, the model also generates a
number of implications for size distribution of firms in different sectors, and
how they are affected by relaxation in barriers to trade and offshoring. We
hope these can be subject to empirical testing in future research.
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42
Appendix
Proof of Proposition 2
Step 1
C
(i)If κ ≥ 1, d[γL /γC ]/dγC = d[λ(γC ; pL , A
)/γC ]/dγC > 0 for any γC so that
AL
AC
λ(γC ; pL , AL )/γC ≥ 1
C
)/γC ]/dγC > 0 if and only if
(ii)If κ < 1, d[γL /γC ]/dγC = d[λ(γC ; pL , A
AL
1
pL < AC /AL (and equivalently γL /γC < ( kkCL ) 1−κ ).
Proof of Step 1
From (7),
d[γL /γC ]/dγC = (1/γC )[
γL +
L
θN
θaL
γC +
C
θN
θaC
−
γL
],
γC
which means that d[γL /γC ]/dγC > 0 if and only if
γL θaL (γL )
γC θaC (γC )
<
.
L
C
θN
(γL )
θN
(γC )
Under this production function in the proposition,
θai (γi )
= (γi )−κ ki
i
θN (γi )
and
pL =
L
1
AC θN
(γL ) + γL θaL (γL )
AC kL γL1−κ + 1 1−κ
=
[
] .
1−κ
C
C
AL θN (γC ) + γC θa (γC )
A L k C γC + 1
In the case of κ ≥ 1 and γL ≥ γC ,
γL θaL (γL )
γC θaC (γC )
1−κ
1−κ
=
(γ
)
k
<
(γ
)
k
=
L
L
C
C
L
C
θN
(γL )
θN
(γC )
implying d[γL /γC ]/dγC > 0. In the case of κ < 1,
γC θaC (γC )
γL θaL (γL )
<
,
L
C
θN
(γL )
θN
(γC )
1
if and only if γL /γC < ( kkCL ) 1−κ which is equivalent to pL < AC /AL .
43
Step 2
k
(i) If κ ≥ 1 −
log[ kC ]
L
m
log[ m L
C
]
,
d[λ(γC ; pL ,
AC
)/γC ]/dγC > 0
AL
holds for pL ∈ [p2L , p1L ).
k
(ii) If 0 ≤ κ < 1 −
log[ kC ]
L
m
log[ m L ]
, for any pL < AC /AL ,
C
d[λ(γC ; pL ,
AC
)/γC ]/dγC > 0
AL
and for any pL > AC /AL ,
d[λ(γC ; pL ,
AC
)/γC ]/dγC < 0.
AL
Proof of Step 2
First suppose that
k
1>κ≥1−
log[ kC ]
L
m
log[ m L
C
]
mL
mC
≤ [ kkCL ]1/(1−κ) and κ < 1, which are equivalent to
. If pL ∈ (p2L , p1L ), since mL /mC > γL /γC ≥ 1 is satisfied
in an equilibrium of supply-side, it implies γL /γC < ( kkCL )1/(1−κ) (or pL <
From (ii) of Step 1, this means that
d[λ(γC ; pL ,
AC
).
AL
AC
)/γC ]/dγC > 0
AL
holds for pL ∈ [p2L , p1L ). From (i) of Step 1, this inequality also holds for
κ ≥ 1. This completes the proof of (i).
k
Next take 0 ≤ κ < 1 −
log[ kC ]
L
m
log[ m L
C
]
. From (ii) in Step 1, for any pL < AC /AL ,
d[λ(γC ; pL ,
AC
)/γC ]/dγC > 0
AL
and for any pL > AC /AL ,
d[λ(γC ; pL ,
AC
)/γC ]/dγC < 0
AL
This completes the proof of (ii).
44
Step 3
Taking pL ∈ (p1L , p2L ) as given, let’s consider the effect of µ on
XL /XC =
AL θaC (γC )
d(aL )
L
AC θa (γL ) [d(aC ) − d(aL )]
We can use the following relationship.
θaC (γC )
]/dµ
θaL (γL )
0
0
θaC (γC ) θaC (γC ) θaL (γL )
AC
= L
)]dγC /dµ
[ C
− L
λ1 (γC ; pL ,
θa (γL ) θa (γC )
θa (γL )
AL
d[
0
0
θC (γC ) θaL (γL ) γL
= aL
[
θa (γL ) θaL (γL ) γC
γC θaC (γC )
θaC (γC )
γL θaL0 (γL )
θaL (γL )
− λ1 (γC ; pL ,
0
A
θaC (γC ) θaL (γL ) λ(γC ; pL , ACL )
= L
[
θa (γL ) θaL (γL )
γC
γL θaL (γL )
L (γ )
θN
L
+1
γC θaC (γC )
C (γ )
θN
C
+1
− λ1 (γC ; pL ,
AC
)]dγC /dµ
AL
A
0
<
AC
)]dγC /dµ
AL
AC
θaC (γC ) θaL (γL ) λ(γC ; pL , ACL )
−
λ
(γ
;
p
,
)]dγC /dµ < 0
[
1
C
L
θaL (γL ) θaL (γL )
γC
AL
C
if d[λ(γC ; pL , A
)/γC ]/dγC > 0. This relationship is using the fact that
AL
0
γi θai
γi θai
=
−κ/(
+ 1)
i
θai
θN
C
and d[λ(γC ; pL , A
)/γC ]/dγC > 0 if and only if
AL
we obtain
γL θaL (γL )
L (γ )
θN
L
<
γC θaC (γC )
C (γ ) .
θN
C
Similarly,
d(aC )
)/dµ
d(aL )
d(aC ) (aC )2 g(aC )/γC
(aL )2 g(aL )/γL
=
[
−
λ1 (γC , pL )]dγC /dµ
d(aL )
d(aC )
d(aL )
A
AC
d(aC ) (aL )2 g(aL )/γL λ(γC ; pL , ACL )
[
− λ1 (γC ; pL ,
)]dγC /dµ > 0
>
d(aL )
d(aL )
γC
AL
d(
45
C
if d[λ(γC ; pL , A
)/γC ]/dγC > 0. This is using the assumption that
AL
increasing in a. This implies that
a2 g(a)
d(a)
is
d(XL /XC )/dµ < 0.
k
for pL ∈
[p2L , p1L )
if κ ≥ 1 −
L
m
log[ m L ]
and for pL ∈ [p2L , AC /AL ) if 0 ≤ κ <
C
k
1−
log[ kC ]
log[ kC ]
L
m
log[ m L ]
.
C
Step 4
Next suppose pL ∈ (p2L , p3L ). γL = γC = γ ∗ is determined by
pL = (
L
AC θN
(γ ∗ ) + γ ∗ θaL (γ ∗ )
) C ∗
.
AL θN (γ ) + γ ∗ θaC (γ ∗ )
γ ∗ is independent of µ. This means that dγ ∗ /dµ = 0. We have only the
direct effect of µ on XL /XC , which is negative.
From step 3 and this step, this completes the proof of (i) and the first
half of (ii) in the proposition.
Step 5
Finally let us show the last part of (ii). Suppose that there does not exist
p̄L ∈ (AC /AL , p1L ) so that XL /XC is increasing in µ for any p ∈ (p̄L , p1L ).
Then p1L has to be non-decreasing in µ. However
dp1L /dµ
=
mL 1 L mL 1
γ θ ( γ )
γC1 θaC (γC1 )
mC C a mC C
1
1
−
]dγC1 /dµ
(pL /γC )[ L mL 1
mL 1 L mL 1
C
1
1 C
1
θN ( mC γC ) + m
γ
θ
(
γ
)
θ
(γ
)
+
γ
θ
(γ
)
N
C
C a
C
C C a mC C
is negative from step 2(ii). This is the contradiction.
Proof of Proposition 6
Let the indirect utility function for country j be V (pjL , I j ) where national
income
I j = pjL XLj + XCj .
46
V (pj ,I j )
By Roy’s identity DLj = − Vp (pjL ,I j ) ,
I
L
dV (pjL , I j )/dpjL =
∂V (pjL , I j ) ∂V (pjL , I j ) dI j
+
∂I j
∂pjL
dpjL
=
dI j
∂V (pjL , I j )
j
[−D
+
]
L
∂I j
dpjL
=
∂V (pjL , I j )
[(XLj − DLj ) + pjL dXLj /dpjL + dXCj /dpjL ]
∂I j
∂V (pj ,I j )
L
Since
> 0, the sign of the welfare effect of the change in pjL is the
∂I j
same as the sign of
(XLj − DLj ) + pjL dXLj /dpjL + dXCj /dpjL
Owing to linear homogeneity of the production function and the common
NL /a chosen by all entrepreneurs in a given sector, XLj = AL FL (NLj , HLj )
where NLj (resp. HLj ) is the total amount of unskilled (resp. skilled) labor used in the L sector of country j. Since pjL dXLj /dNLj = wj and γLj =
j
j
j
∂FL (NL
,HL
)/∂HL
j
j
j
∂FL (NL
,HL
)/∂NL
,
pjL dXLj /dpjL = pjL AL [∂FL (NLj , HLj )/∂NLj [dNLj /dpjL ] + ∂FL (NLj , HLj )/∂HLj [dHLj /dpjL ]]
= wj [dNLj /dpjL + γLj dHLj /dpjL ]
Similarly
dXCj /dpjL = AC [∂FC (NCj , HCj )/∂NCj [dNCj /dpjL ] + ∂FC (NCj , HCj )/∂HCj [dHCj /dpjL ]]
= wj [dNCj /dpjL + γCj dHCj /dpjL ]
Therefore
pjL dXLj /dpjL +dXCj /dpjL = wj [d(NLj +NCj )/dpjL +γCj d(HLj +HCj )/dpjL +(γLj −γCj )dHLj /dpjL ].
j
j
j3 j2
j
j1
Since HLj = µd(ajL ) for pjL ∈ (pj2
L , pL ) and γL = γC for pL ∈ (pL , pL ),
(γLj − γCj )dHLj /dpjL = µ(γLj − γCj )d(d(ajL ))/dpjL
j1
j
j
j
j
j
holds for pjL ∈ (pj3
L , pL ). With NL + NC = µG(aC ) + (1 − µ) and HL + HC =
j
j
j3 j1
µd(aC ) for pL ∈ (pL , pL ),
(XLj − DLj ) + pjL dXLj /dpjL + dXCj /dpjL
= (XLj − DLj ) + wj µ[dG(ajC )/dpjL + γCj d(d(ajC ))/dpjL + (γLj − γCj )d(d(ajL ))/dpjL ]
= (XLj − DLj ) + wj µ[(1 − mC )g(ajC )dajC /dpjL − (γLj − γCj )ajL g(ajL )dajL /dpjL ]
47